The Mathematics of It All

If you are anything like me, you struggled with math in school. Oh, I could write like Faulkner or Hemingway, play the saxophone like Jimmy Dorsey, draw like Rodin or Picasso….but when it came to numbers, I was lost.

Eventually, with the help of a very good teacher, I caught on to basic algebra in college. Later, when I started playing with CAD and 3d design applications, I started getting good at geometry. But here’s the thing that I really started to appreciate: there is a lot of mathematics contained in music.

It’s something that good percussionists understand from the get-go (when hopefully they master the rudiments), but those who sing, play tonal instruments or compose or arrange music don’t always realize it. Interestingly, in  the education system of ancient Greece and Rome, music and mathematics were not separate subject areas; music was considered part of the mathematics curriculum.

Why am I going into this? Because in its most basic form, mathematics is simply organized patterns – like great compositions, orchestrations and improvisations. Once you learn to recognize and employ patterns, you can cross the line from being average or simply “good” into the realm of excellence.

In my last post, I started discussing Jerome Kern’s All The Things You Are, a song that has a unique chord progression (well, almost: the first five measures of Fly Me To The Moon are very similar) as well as an exquisite construction based on patterns.  In music however, we call these patterns motives.

Disclaimer: the following musical examples do not necessarily reflect Kern’s original score – and I’ve kept them deliberately simple for illustrative purposes.

Let’s start with the verse.  On the surface, there’s nothing too remarkable about it. However, it clearly demonstrates Kern’s formal musical training at the New York College of Music (present-day NYU Steinhardt):

Note how this motive is developed, with the eighth note pattern in beats 2-4  rising a major second each time (in musical parlance, this is known as a sequence). Also, pay attention to the circled notes. These are what Schenker would have identified as the primary tones, the objectives toward which the preceding notes are leading. They actually make for a very nice guide tone line, which could be the basis of an improvisation or a string or reed section accompaniment.

It’s just a fragment of a G major scale – nothing too exciting or elaborate – but it’s an excellent place to start for those new to jazz improvisation and/or  orchestration and arranging.

Kern maintains the pattern in the next two measures before ending this part of his tune – except this time,  the second permutation drops the pattern a minor third:

And of course, we can add our guide tone line in our “string section”  (listen to how it divides toward the end of this selection):

Finally, as Kern brings the verse to a close, he continues with a development of his initial motivic idea:

BTW, despite the simple musical accompaniment I’ve provided here, when I’m performing this song, I usually put a sustained D pedal tone in the bass  under the first and second measures of the above example – and would probably make this and other choices if I was doing a professional arrangement – for reasons (beyond the present topic)  I’ll discuss in a future post. (Actually, your ears are probably telling you that it sounds appropriate – but you may not understand just why.)

In the next post, we will see how Kern uses his initial two-note motive of a rising fourth (from the dominant to the tonic) as the basis for the entire refrain, creating exquisite unity between verse and “burthen” (as Kern referred to it).

Incidentally, did anyone notice my use of first species counterpoint and contrary motion in any of the sound examples above?

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About KJ at Guide Tone Lines

KJ McElrath earned his Masters degree in Music Theory and Composition from Central Washington University. Composer of several works for big band, wind ensemble and orchestra, which can be heard at I perform a cabaret act with Athena McElrath as McElrath Cabaret, which you can find at
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3 Responses to The Mathematics of It All

  1. AM says:

    I actually kind of understood that, and I’m a singer!

    • And this kind of knowledge is even more important for singers and vocal arrangers. Because singing depedns on aural skills to a much greater degree than playing most instruments, it’s important that vocalists in an ensemble situation are able to hear and make sense of these things.

  2. Pingback: Finding The Guide Tone Lines: “All The Things You Are” | Guide Tone Lines